MANIFOLDS OF DIFFERENTIABLE DENSITIES

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class C-b(k) with respect to appropriate reference measures. The case k = infinity, in which the manifolds are modelled on Frechet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari's alpha-covariant derivatives for all alpha is an element of R. By construction, they are C-infinity - embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (alpha = +/- 1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the alpha-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the alpha-divergences are of class C-infinity.